3.981 \(\int \frac{x^2}{\sqrt{3-2 x^4}} \, dx\)

Optimal. Leaf size=48 \[ \frac{\sqrt [4]{3} E\left (\left .\sin ^{-1}\left (\sqrt [4]{\frac{2}{3}} x\right )\right |-1\right )}{2^{3/4}}-\frac{\sqrt [4]{3} F\left (\left .\sin ^{-1}\left (\sqrt [4]{\frac{2}{3}} x\right )\right |-1\right )}{2^{3/4}} \]

[Out]

(3^(1/4)*EllipticE[ArcSin[(2/3)^(1/4)*x], -1])/2^(3/4) - (3^(1/4)*EllipticF[ArcS
in[(2/3)^(1/4)*x], -1])/2^(3/4)

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Rubi [A]  time = 0.124061, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{\sqrt [4]{3} E\left (\left .\sin ^{-1}\left (\sqrt [4]{\frac{2}{3}} x\right )\right |-1\right )}{2^{3/4}}-\frac{\sqrt [4]{3} F\left (\left .\sin ^{-1}\left (\sqrt [4]{\frac{2}{3}} x\right )\right |-1\right )}{2^{3/4}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/Sqrt[3 - 2*x^4],x]

[Out]

(3^(1/4)*EllipticE[ArcSin[(2/3)^(1/4)*x], -1])/2^(3/4) - (3^(1/4)*EllipticF[ArcS
in[(2/3)^(1/4)*x], -1])/2^(3/4)

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Rubi in Sympy [A]  time = 13.7575, size = 49, normalized size = 1.02 \[ \frac{\sqrt [4]{6} E\left (\operatorname{asin}{\left (\frac{\sqrt [4]{2} \cdot 3^{\frac{3}{4}} x}{3} \right )}\middle | -1\right )}{2} - \frac{\sqrt [4]{6} F\left (\operatorname{asin}{\left (\frac{\sqrt [4]{2} \cdot 3^{\frac{3}{4}} x}{3} \right )}\middle | -1\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(-2*x**4+3)**(1/2),x)

[Out]

6**(1/4)*elliptic_e(asin(2**(1/4)*3**(3/4)*x/3), -1)/2 - 6**(1/4)*elliptic_f(asi
n(2**(1/4)*3**(3/4)*x/3), -1)/2

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Mathematica [A]  time = 0.0521902, size = 38, normalized size = 0.79 \[ \frac{\sqrt [4]{3} \left (E\left (\left .\sin ^{-1}\left (\sqrt [4]{\frac{2}{3}} x\right )\right |-1\right )-F\left (\left .\sin ^{-1}\left (\sqrt [4]{\frac{2}{3}} x\right )\right |-1\right )\right )}{2^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/Sqrt[3 - 2*x^4],x]

[Out]

(3^(1/4)*(EllipticE[ArcSin[(2/3)^(1/4)*x], -1] - EllipticF[ArcSin[(2/3)^(1/4)*x]
, -1]))/2^(3/4)

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Maple [A]  time = 0.082, size = 69, normalized size = 1.4 \[ -{\frac{\sqrt{3}\sqrt [4]{6}}{18}\sqrt{9-3\,{x}^{2}\sqrt{6}}\sqrt{9+3\,{x}^{2}\sqrt{6}} \left ({\it EllipticF} \left ({\frac{x\sqrt{3}\sqrt [4]{6}}{3}},i \right ) -{\it EllipticE} \left ({\frac{x\sqrt{3}\sqrt [4]{6}}{3}},i \right ) \right ){\frac{1}{\sqrt{-2\,{x}^{4}+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(-2*x^4+3)^(1/2),x)

[Out]

-1/18*3^(1/2)*6^(1/4)*(9-3*x^2*6^(1/2))^(1/2)*(9+3*x^2*6^(1/2))^(1/2)/(-2*x^4+3)
^(1/2)*(EllipticF(1/3*x*3^(1/2)*6^(1/4),I)-EllipticE(1/3*x*3^(1/2)*6^(1/4),I))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{-2 \, x^{4} + 3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/sqrt(-2*x^4 + 3),x, algorithm="maxima")

[Out]

integrate(x^2/sqrt(-2*x^4 + 3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{2}}{\sqrt{-2 \, x^{4} + 3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/sqrt(-2*x^4 + 3),x, algorithm="fricas")

[Out]

integral(x^2/sqrt(-2*x^4 + 3), x)

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Sympy [A]  time = 1.92152, size = 39, normalized size = 0.81 \[ \frac{\sqrt{3} x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{2 x^{4} e^{2 i \pi }}{3}} \right )}}{12 \Gamma \left (\frac{7}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(-2*x**4+3)**(1/2),x)

[Out]

sqrt(3)*x**3*gamma(3/4)*hyper((1/2, 3/4), (7/4,), 2*x**4*exp_polar(2*I*pi)/3)/(1
2*gamma(7/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{-2 \, x^{4} + 3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/sqrt(-2*x^4 + 3),x, algorithm="giac")

[Out]

integrate(x^2/sqrt(-2*x^4 + 3), x)